Sensor mission assignment involves matching the sensing resources of a wireless sensor network (WSN) to appropriate tasks (missions), which may come to the network dynamically. Although solutions for WSNs with battery-operated nodes have been proposed for this problem, no attention has been given to networks whose nodes have energy-harvesting capabilities and are powered in part by uncontrollable environmental sources, which impose quite a different energy model. In this article we address this problem by providing both an analytical model and a distributed heuristic, called EN-MASSE, specifically tailored for energy-harvesting mission-centric WSNs. To assess the performance of our proposed solution we have interfaced TelosB nodes with solar cells and performed extensive experiments to derive models and traces of solar energy acquisition. We use such real-life traces in our simulations. A comparative performance evaluation between EN-MASSE and other schemes previously proposed in the literature has shown that our solution significantly outperforms existing energy-harvesting-unaware mission assignment schemes. Moreover, using our analytical model as a benchmark, we also show that the profit earned by EN-MASSE is close to the optimum. Finally, we have implemented our proposed solution in TinyOS and experimentally validated its performance, showing the effectiveness of our approach.